I concluded my series of talks by showing the following theorem of Viale:

Theorem (Viale). Assume and let be an inner model where is regular and such that . Then .

This allows us to conclude, via the results shown last time, that if holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .

An elaboration of this argument is expected to show that, at least if we strengthen the assumption of to , then computes correctly ordinals of cofinality .

Under an additional assumption, Viale has shown this: If holds in , is a strong limit cardinal, , and in we have that is regular, then in , the cofinality of cannot be . The new assumption on allows us to use a result of Dzamonja and Shelah, On squares, outside guessing of clubs and , Fund. Math. 148 (1995), 165-198, in place of the structure imposed by . It is still open if the corresponding covering statement follows from , which would eliminate the need for this the strong limit requirement.

I presented a sketch of a nice proof due to Todorcevic that implies the P-ideal dichotomy . I then introduced Viale’s covering property and showed that it follows from . Next time I will indicate how it can be used to provide a proof of part 1 of the following theorem:

Theorem (Viale). Assume is an inner model.

If holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .

If holds in , is a strong limit cardinal, , and in we have that is regular, then in , the cofinality of cannot be .

It follows from this result and the last theorem from last time that if is a model of and a forcing extension of an inner model by a cardinal preserving forcing, then .

In fact, the argument from last time shows that we can weaken the assumption that is a forcing extension to the assumption that for all there is a regular cardinal such that in we have a partition where each is stationary in .

It is possible that this assumption actually follows from in . However, something is required for it: In Gitik, Neeman, Sinapova, A cardinal preserving extension making the set of points of countable cofinality nonstationary, Archive for Mathematical Logic, vol. 46 (2007), 451-456, it is shown that (assuming large cardinals) one can find a (proper class) forcing extension of that preserves cardinals, does not add reals, and (for some cardinal ) the set of points of countable -cofinality in is nonstationary for every regular . Obviously, this situation is incompatible with in , by Viale’s result.

Remark 1. is club in , so any is stationary as a subset of iff it is stationary as a subset of . It follows that proper forcing preserves stationary subsets of .

Remark 2. Proper forcing extensions satisfy the countable covering property with respect to , namely, if is proper, then any countable set of ordinals in is contained in a countable set of ordinals in . We won’t prove this for now, but the argument for Axiom A posets resembles the general case reasonably:

Given a name for a countable set of ordinals in the extension, find an appropriate regular and consider a countable elementary containing , , and any other relevant parameters. One can then produce a sequence such that

Each is in .

.

, where enumerates the dense subsets of in .

Let for all . Then , so is a countable set of ordinals in containing in . A density argument completes the proof.

Woodin calls a poset weakly proper if the countable covering property holds between and . Not every weakly proper forcing is proper, but the countable covering property is a very useful consequence of properness. On the other hand, standard examples of stationary set preserving posets that are not proper, like Prikry forcing (changing the cofinality of a measurable cardinal to without adding bounded subsets of ) or Namba forcing (changing the cofinality of to while preserving are not weakly proper, and account for some of the usefulness of over .

The following is obvious:

Fact. Assume is weakly proper. Then either adds no new -sequences of ordinals, or else it adds a real.

The relation between the reals and the -sequences of ordinals in the presence of strong forcing axioms like is a common theme I am exploring through these talks.

I’ll use this post to provide some notes about consistency strength of the different natural hierarchies that forcing axioms and their bounded versions suggest. This entry will be updated with some frequency until I more or less feel I don’t have more to add. Feel free to email me additions, suggestions and corrections, or to post them in the comments. In fact, please do.

Today I started a series of talks on “Forcing axioms and inner models” in the Set Theory Seminar. The goal is to discuss a few results about strong forcing axioms and to see how these axioms impose a certain kind of `rigidity’ to the universe.

I motivated forcing axioms as trying to capture the intuition that the universe is `wide’ or `saturated’ in some sense, the next natural step after the formalization via large cardinal axioms of the intuition that the universe is `tall.’

The extensions of obtained via large cardinals and those obtained via forcing axioms share a few common features that seem to indicate their adoption is not arbitrary. They provide us with reflection principles (typically, at the level of the large cardinals themselves, or at small cardinals, respectively), with regularity properties (and determinacy) for many pointclasses of reals, and with generic absoluteness principles.

The specific format I’m concentrating on is of axioms of the form for a class of posets, stating that any admits filters meeting any given collection of many dense subsets of . The proper forcing axiom is of this kind, with being the class of proper posets. The strongest axiom to fall under this setting is Martin’s maximum , that has as the class of all posets preserving stationary subsets of .

Of particular interest is the `bounded’ version of these axioms, which, if posets in preserve , was shown by Bagaria to correspond precisely to an absoluteness statement, namely that for any .

In the next meeting I will review the notion of properness, and discuss some consequences of .

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

The usual definition of a series of nonnegative terms is as the supremum of the sums over finite subsets of the index set, $$\sum_{i\in I} x_i=\sup\biggl\{\sum_{j\in J}x_j:J\subseteq I\mbox{ is finite}\biggr\}.$$ (Note this definition does not quite work in general for series of positive and negative terms.) The point then is that is $a< x

The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]

Given a class $S$, to say that it can be proper means that it is consistent (with the axioms under consideration) that $S$ is a proper class, that is, there is a model $M$ of these axioms such that the interpretation $S^M$ of $S$ in $M$ is a proper class in the sense of $M$. It does not mean that $S$ is always a proper class. In fact, it could also be consis […]

As the other answers point out, the question is imprecise because of its use of the undefined notion of "the standard model" of set theory. Indeed, if I were to encounter this phrase, I would think of two possible interpretations: The author actually meant "the minimal standard model of set theory", that is, $L_\Omega$ where $\Omega$ is e […]